The stable blow-up conjecture for the $L^2$-critical generalized Hartree equation

Let $u$ be a stable blow-up solution of the $L^2$-critical generalized Hartree equation, with blow-up time $T$, spatial dimension $d$, scale $L(t)$, center $x(t)$, phase parameter $\gamma(t)$, and profile $Q$ a ground state solution of the profile equation. $L^2$-critical generalized Hartree conjecture. A stable blow-up solution has a self-similar structure and satisfies

$$\lim_{t \rightarrow T}\\|\nabla u(\cdot,t) \\|_{L^2_x} =\left( \frac{\ln|\ln(T-t)|}{2\pi(T-t)} \right)^{\frac{1}{2}} \quad \text{as} \quad {t \to T},$$

with profile asymptotic to

$$u(x,t) \sim \dfrac{1}{L(t)^{\frac{d}{2}}} Q\left(\frac{x-x(t)}{L(t)}\right) e^{i\gamma(t)}.$$

This is the logarithmic-logarithmic blow-up rate, and the conjectured dynamics is analogous to stable blow-up in the $L^2$-critical nonlinear Schrödinger equation.